On the Applications of the Special Class of Atomic Functions: Practical Aspects and Perspectives

A special class of atomic functions, which are compactly supported solutions of linear functional differential equations with constant coefficients and linear transformations of the argument, and their basic properties are considered. Spaces of linear combinations of shifts of these functions are given. Such spaces have good approximation properties and combine smoothness of elements with an existence of compactly supported basis. For this reason, their application to solution of different problems is perspective. System of atomic wavelets, which is a basis of such spaces, is a practical tool for data analysis and processing. The aim of this paper is to analyze atomic wavelets expansion algorithm in terms of complexity. Using properties of atomic functions and atomic wavelets, it is shown that expansion algorithm has linear time and memory complexity. Moreover, the numerical scheme can be easily parallelized. This means that application of atomic wavelets is promising especially when processing digital images and video.